Why do some bones in an animation lose their position when applying scale to their armature?
Generally if an armature already has animation data, it's not safe to scale it and apply the transformation. It has something to do with changing the armature object's transformation matrix, because each keyframes in an action for a bone is recorded and evaluated against it.

To illustrate, here I have a keyframed armature, and I just add a free-floating bone parented to torso:



If I halve the armature's size in object mode, each bone's matrix stays the same:



But after I apply the scale transformation, it's a different matrix for each bones:



Whatever matrix operation is applied against the keyframe data, no longer turns out the correct transformation. In my experience, only free-floating bones are prone to this problem (the feet doesn't stray like this when animated in FK).